**Question:**

Find the relationship between '*a*' and '*b*' so that the function '*f*' defined by

$f(x)= \begin{cases}a x+1, & \text { if } x \leq 3 \\ b x+3, & \text { if } x \geq 3\end{cases}$

is continuous at *x* = 3.

**Solution:**

Given: $f(x)=\left\{\begin{array}{l}a x+1, \text { if } x \leq 3 \\ b x+3, \text { if } x>3\end{array}\right.$

We have

(LHL at $x=3$ ) $=\lim _{x \rightarrow 3^{-}} f(x)=\lim _{h \rightarrow 0} f(3-h)=\lim _{h \rightarrow 0} a(3-h)+1=3 a+1$

(RHL at $x=3$ ) $=\lim _{x \rightarrow 3^{+}} f(x)=\lim _{h \rightarrow 0} f(3+h)=\lim _{h \rightarrow 0} b(3+h)+3=3 b+3$

If $f(x)$ is continuous at $x=3$, then

$\lim _{x \rightarrow 3^{-}} f(x)=\lim _{x \rightarrow 3^{+}} f(x)$

$\Rightarrow 3 a+1=3 b+3$

$\Rightarrow 3 a-3 b=2$

Hence, the required relationship between $a \& b$ is $3 a-3 b=2$.